# Exploring the growth value equity valuation model with data visualization

## Abstract

The Growth Value Model (GVM) proposed theoretical closed form formulas consisting of Return on Equity (ROE) and the Price-to-Book value ratio (P/B) for fair stock prices and expected rates of return. Although regression analysis can be employed to verify these theoretical closed form formulas, they cannot be explored by classical quintile or decile sorting approaches with intuition due to the essence of multi-factors and dynamical processes. This article uses visualization techniques to help intuitively explore GVM. The discerning findings and contributions of this paper is that we put forward the concept of the smart frontier, which can be regarded as the reasonable lower limit of P/B at a specific ROE by exploring fair P/B with ROE-P/B 2D dynamical process visualization. The coefficients in the formula can be determined by the quantile regression analysis with market data. The moving paths of the ROE and P/B in the current quarter and the subsequent quarters show that the portfolios at the lower right of the curve approaches this curve and stagnates here after the portfolios are formed. Furthermore, exploring expected rates of return with ROE-P/B-Return 3D dynamical process visualization, the results show that the data outside of the lower right edge of the “smart frontier” has positive quarterly return rates not only in the t + 1 quarter but also in the t + 2 quarter. The farther away the data in the t quarter is from the “smart frontier”, the larger the return rates in the t + 1 and t + 2 quarter.

## Introduction

The estimation of expected rates of return on stocks is the core of stock investment. The valuation of stock prices is the core of the estimation of expected rates of return on stocks. The valuation of stock prices could be based on two claims of the shareholders (Ohlson 2001; Celik 2012; Richardson and Tinaikar 2004; Yeh and Hsu 2011; Lundholm and Sloan 2012; Rossi 2016; Ho et al. 2017):

1. (1)

The right to request net assets of the company in liquidation. It is the rationale of asset-based approach (Gilson et al. 2000; Garza-Gomez 2001).

2. (2)

The right to request dividends during operation. It is the rationale of income-based approach (Francis et al. 2000; Adsera and Vinolas 2003; Fernández 2007; Sweeney 2014).

In addition to the asset-based and the income-based approaches, the market-based approach is another option. The market-based approach depicts that companies with similar market scales, technologies, products, or customers are suggested to have similar performance levels and ratios, such as the price-to-earnings ratio (P/E) or the price-to-book ratio (P/B or PBR) (Lie and Lie 2002; Liu et al. 2002; Park and Lee 2003). Hence, the value of a company can be inferred from the value of similar companies. For example, the stock price of a company can be estimated by multiplying earnings per share (or book value per share) with the average P/E (or P/B) of its similar companies.

There are pros and cons of different stock valuation approaches. Although the income-based and the asset-based methods have more solid theoretical basis, either one of these two only focuses on the company’s income or its assets, and they may not be able to fit the market data. On the other hand, the market-based approach can fit the market data but it lacks a sound theoretical foundation.

To overcome the shortcomings of the traditional valuation models aforementioned, some hybrid approaches have been proposed. For example, Growth Value Model (GVM) is an equity-valuation method that formally combines the essence of the asset-based approach and the income-based approach (Yeh and Hsu 2011; Liu and Yeh 2014; Yeh and Lien 2017; Kong et al. 2019). Current book value per share is not a part of the stock's intrinsic value. It is the initial value of the stock's intrinsic value. The initial value will grow through return on equity (ROE) with a mean reversion trajectory. The fair stock price is the discounted value of the future shareholders’ equity at time infinity. Thus, a theoretical closed form formula consists of the growth coefficient and the required rate of return of equity capital. It can be derived through the discount process as follows (Yeh and Lien 2017; Kong et al. 2019)

$$P_{ \, 0} = B_{0} \cdot \left( {\frac{{1 + ROE_{0} }}{1 + r}} \right)^{m}$$
(1)

where $$P_{ \, 0}$$ is the fair stock price; namely, intrinsic value. $$B_{ \, 0}$$ is the current book value per share. $$ROE_{ \, 0}$$ is the current return on equity. $$r$$ is the required rate of return of equity capital. m is the growth coefficient that can be estimated with market data. The greater the growth coefficient, the greater the influence of ROE on stock price.

Moreover, based on GVM, a theoretical closed form formula can be developed to estimate the expected rates of return of equity securities.

$$R = \lambda \cdot \left( {\frac{{B_{0} }}{{P_{0} }}} \right)^{b} \cdot \left( {1 + ROE_{0} } \right)^{c} - 1$$
(2)

where $$P_{ \, 0}$$ is the current stock price; coefficients $$\lambda ,b,c$$ can be determined by regression analysis and market data.

Although we can employ the regression analysis to verify these theoretical closed form formulas, they cannot be explored by classical quintile or decile sorting approaches with intuition due to the essence of multi-factors and dynamical processes.

The essence of quantitative investment is the multi-factor model, which generates enormous quantities of factor data. Therefore, this makes well-experienced portfolio managers find it difficult to navigate. This has led to portfolio analysis and factor research being limited by a lack of intuitive visual analytics tools (Yue et al. 2019).

Data visualization is the graphical representation of information and data. Our eyes are drawn to colors and patterns. We can quickly identify blue from red, circle from square. The visual culture includes everything from art to science. Data visualization is regarded as a modern concept by many disciplines, which has the same meaning as visual communication. It involves producing images that communicate in relationships among the represented data and viewers of the images. This communication is achieved through the use of a systematic mapping between data values and graphic marks in the creation of the visualization. Dots, lines, or bars can be used to encode digital data to visually convey quantitative information. This mapping establishes a visual representation of data values. It also determines how and to what extent a property of a graphic mark, such as size or color, will evolve to reflect changes in the value of a datum. Effective visualization can help users analyze data and obtain evidence. It makes complex information easier to understand and use (Telea 2014; Chen et al. 2007).

Some literature explored stock data with data visualization (Liu and Yeh 2014; Yeh and Hsu 2014; Rea and Rea 2014; Yue et al. 2019). For example, Yeh and Hsu (2014) used time series data to visualize the change of stock prices and stock returns for value stocks and growth stocks before and after the formation of the portfolios. They also established a dynamic model showing the returns from value stocks and growth stocks, which is called the exponential decay model. Their results discover that value stocks exhibit a significantly over-reacting phenomenon, while growth stocks exhibit an obviously under-reacting phenomenon.

In this paper, visualization techniques are applied to intuitively explore the GVM. The research samples are the stocks of all listed companies in the Taiwan stock exchange. The study period is from 2008/Q1 to 2018/Q4 with 11 years (44 quarters). There are more than 950 listed companies in Taiwan Stock Exchange. Its total market capitalization is currently about US$2100 billion, and the average daily trading value is currently about US$12 billion on 14th January, 2022. We put forward the concept of smart frontier by exploring fair P/B with ROE-P/B 2D dynamical process visualization. This can be regarded as the reasonable lower limit of P/B at a specific ROE. Furthermore, exploring expected rates of return with ROE-P/B-Return 3D dynamical process visualization, we found that the stock outside the lower right edge of “smart frontier” has a positive quarterly return rate not only in the t + 1 quarter but also in the t + 2 quarter. The farther away the data in the t quarter is from the “smart frontier”, the larger the return rates in the t + 1 and t + 2 quarter.

There are two discerning findings and contributions of this paper while the research results and theoretical models were applied to obtain the fair price of stocks in this paper. First, according to findings from data visualization, we proposed the novel concept of smart frontiers, that is, the essence of the company (ROE) and the valuation of the market (P/B) have a curvilinear relationship. There are few investment portfolios at the bottom right of this curve. Thus, the smart frontier can be regarded as the reasonable lower bound of P/B. This is because the P/B of the portfolio below this curve is lower than the lower bound of reasonable P/B consistent with the valuation of the company's essence (ROE), and the stock price is undervalued. Therefore, it implies that smart investors should purchase these undervalued stocks until the stock prices rise and return to the boundary.

Second, the smart frontier can be established by the equation derived from the Growth Value Model. To determine the coefficients in the model by regression analysis and market data, the market Price-to-Book value ratio is regarded as the dependent variable, and Return on Equity is regarded as the independent variable. However, since smart frontier is the reasonable lower bound of P/B in regards to a specific ROE, the classical regression analysis, which uses only the mean value as the dependent variable, cannot obtain the lower bound relationship curve. Quantile regression analysis has been widely adopted in various research fields in recent years because it adopts the median or another quantile values as the dependent variables of the regression models (Koenker and Hallock 2001; Mello and Perrelli 2003; Gowlland et al. 2009; Liao and Wang 2012; Uribe and Guillen 2020). For instance, it constructs a series of regression models for various quantiles such as 1/10, 2/10,…, 9/10. To find the smart frontier, we set the θ = 0.01. Thus, the probability that the P/B is lower than the predicted value is 0.01.

The second section of this article explains the theoretical model, which can derive the stock price valuation formula and the expected rate of return estimation formula with the Price-to-Book value ratio (P/B) and Return on Equity (ROE) as variables. In the third section, we explain how to explore the dynamic relationship between company essence (ROE) and market valuation (P/B) by using visualization technology. The stock price valuation formula is to be verified as well. The fourth section explains how to use visualization technology to explore the dynamic relationship among ROE, P/B, and stock return. Also, the estimation formula on expected rates of return is to be verified. Section 5 summarizes the results of this paper.

## Growth value model (GVM)

Yeh and Hsu (2011) proposed the Growth Value Model (GVM), which integrates the principles of income-based and asset-based approaches. This model is based on the assumption that Price-to-Book value ratio (P/B) and Return on Equity (ROE) both exhibit mean reversion and the theoretical valuation formula is derived accordingly. Mean reversion is an important concept in finance. It indicates that a variable tends to approach the mean value, regardless of whether the initial value of the variable is higher or lower than the long-term mean value (Chen and Lin 2011; Welc 2011; Canarella et al. 2013; Holland 2018).

Due to the complex process, Yeh and Lien (2017) proposed a simplified version of the previously proposed GVM. It assumes that only ROE exhibits mean reversion in this model and a closed form valuation formula, Eq. (1), is derived accordingly. This model is based on the following three assumptions (Yeh and Lien 2017; Kong, et al. 2019).

### Assumption 1:

Growth rate of shareholders’ equity equals Return on Equity (ROE).

Many companies distribute cash dividends. However, shareholders still can reinvest the distributed dividends into the companies making the dividend policy irrelevant to shareholders’ equity (Modigliani and Miller, 1959; Miller and Modigliani, 1961). Hence, the shareholders’ equity of the n-th period could be estimated by the current shareholders’ equity (B0) and the expected growth rate of shareholders’ equity, i.e., Return on Equity (ROE).

$$B_{n} = B_{0} \cdot \prod\limits_{t = 1}^{n} {\left( {1 + ROE_{t} } \right)}$$
(3)

### Assumption 2:

The fair share price is equal to the discount of shareholders' equity at time infinity.

The fair stock price $$P_{ \, 0}$$ is the discounted value of the future shareholders’ equity at time infinity.

$$P_{0} = \mathop {\lim }\limits_{n \to \infty } \frac{{B_{n} }}{{\left( {1 + r} \right)^{n} }}$$
(4)

r = discount rate, desired rate of return.

### Assumption 3:

The discounted growth rate of shareholders’ equity follows the mean reversion relationship presented in the equation below.

$$\frac{{1 + ROE_{t + 1} }}{1 + r} = \left( {\frac{{1 + ROE_{t} }}{1 + r}} \right)^{a}$$
(5)

where a is the persistence rate of mean reversion and is a constant smaller than 1. The value of this formula approaches 1 as t approaches infinity.

Based on the above three assumptions, a closed form formula can be derived through the calculation of discrete infinite series as follows.

$$P_{0} = B_{0} \cdot \left( {\frac{{1 + ROE_{0} }}{1 + r}} \right)^{{\frac{a}{1 - a}}}$$
(6)

let

$$m = \frac{a}{1 - a}$$
(7)

and

$$k = \left( {\frac{1}{1 + r}} \right)^{{\frac{a}{1 - a}}} = \left( {\frac{1}{1 + r}} \right)^{m}$$
(8)

Then Eq. (6) can be simplified as follows.

$$P_{ \, 0} = k \cdot B_{0} \cdot \left( {1 + ROE_{0} } \right)^{m}$$
(9)

The coefficient $$m$$ is defined as the growth coefficient of shareholders’ equity because it influences the effect of the current ROE on the growth of the fair stock price. The larger the $$m$$ value, the greater the impact of the current ROE on the fair stock price. The coefficients m and k are determined by regression analysis and market data. Thus, Eq. (9) can be rewritten below.

$$P_{ \, 0} /B_{0} = k \cdot \left( {1 + ROE_{0} } \right)^{m}$$
(10)

Then, the fair Price-to-Book value ratio can be regarded as the dependent variable, and the current Return on Equity ($$ROE_{ \, 0}$$) as the independent variable, then the coefficient m and k can be determined by regression analysis and market data.

The expected rate of return can be derived from the following formula if the current stock price is known:

$$R = \lambda \cdot \left( {\frac{{B_{0} }}{{P_{0} }}} \right)^{b} \cdot \left( {1 + ROE_{0} } \right)^{c} - 1$$
(11)

where $$P_{ \, 0}$$ is the current stock price; coefficients $$\lambda ,b,c$$ can be determined by regression analysis and market data.

Because $$B_{0} /P_{0}$$ = 1 means that the stock price equals stockholders’ equity per share, under this condition it is reasonable to infer that the expected return rate is the same as ROE for a long run. Therefore, coefficients λ and c should be close to 1.0 in regression analysis.

## Data visualization

### Method

The visualization method of the two-dimensional dynamic process of the value factor (P/B) and the growth factor (ROE) of stocks can be divided into two parts:

#### Sorting and calculation

1. (1)

First, the stocks are sorted into deciles according to the ROE of each quarter. Each decile is sorted into deciles according to the P/B ratio at the end of the quarter. Then, a total of 100 investment portfolios are obtained (Fig. 1).

2. (2)

Calculate the average value of the quarterly ROE and P/B ratio of all stocks in each portfolio for each quarter.

3. (3)

Calculate the average value of the quarterly ROE and P/B ratio of all stocks in each portfolio in their several subsequent quarters such as four or eight subsequent quarters.

The above process can be a rolling process (Fig. 2). We take the following four quarters as an example to make an observation. If the t quarter (formation period) is Q1 of 2008, the subsequent t + 1 to t + 4 quarters (observation period) are from Q2 of 2008 to Q1 of 2009. If the t quarter is Q2 of 2008, the subsequent t + 1–t + 4 quarters are from Q3 of 2008 to Q2 of 2009. Finally, all the t quarters (formation period) and the subsequent t + 1–t + 4 quarters (observation period) are respectively added up and averaged to obtain a more reliable total average. This rolling process can eliminate the influence of long/short markets. Therefore, the data presents an intrinsic relationship between the company's essence (ROE) and the market valuation (P/B) during the formation period and the observation period.

#### Visual display

1. (1)

These 100 investment portfolios are formed in the t quarter. The average ROE for each quarter is the horizontal axis, and the average P/B ratio is the vertical axis of the two-dimensional graph.

2. (2)

To explore the changes in ROE and P/B ratio of these portfolios in the subsequent quarters, the ROE and P/B average values of these portfolios in the subsequent quarters are also plotted on the graph, connected with straight lines, and marked at the ending points. This is convenient for observers to discover the characteristics and patterns of the dynamic relationship between these two variables.

### Data sources

This research takes the Taiwan stock market as the research scope and uses the Taiwan Economic Journal (TEJ) Database as the data source. The research samples are the stocks of all listed companies. The study period is from 2008/Q1 to 2018/Q4 with 11 years (44 quarters). Figure 3 shows the scatter diagram of the ROE and the P/B ratio. It signifies that the larger the ROE, the higher the P/B.

To explore and verify equity valuation model by applying the data visualization method:

1. (1)

First, the stocks in the t quarter are sorted into deciles according to the ROE of each quarter. Each decile is sorted into deciles according to the P/B ratio at the end of the quarter. Then, 100 investment portfolios are obtained.

2. (2)

Next, we calculate the average value of the quarterly ROE and P/B ratio of all stocks in each portfolio of each quarter and their eight subsequent quarters: t, t + 1, t + 2, t + 3,…, t + 8. For example, the 9 quarters from 2008/Q1 to 2010/Q1 are calculated if t = 2008/Q1.

3. (3)

Finally, all the t quarters (formation period) and the subsequent t + 1–t + 8 quarters (observation period) are added up and averaged respectively in order to eliminate the influence of long/short markets and to obtain a more reliable total average.

### Data visualization: all stocks

#### The ROE and P/B in the current quarter and the subsequent first quarter after the portfolios are formed

The ROE and P/B of the current quarter and the subsequent first quarter after the portfolios being formed are shown in Fig. 4. The circle end of the line segment is the position of the subsequent first quarter. The other end is the position when the portfolio is formed. Some important implications can be found from Fig. 4.

1. (1)

The data points of the subsequent quarters are obviously gathered in the curved area from the lower left to the upper right after the portfolios are formed. This reflects the fact that the market gives higher P/B valuations to stocks with a higher ROE.

2. (2)

They move to the top left in quarter t + 1 if the portfolios formed by sorting in quarter t are at the bottom right of the curved area. This indicates that either the stock’s P/B in the next quarter increases and/or the company’s ROE in the next quarter decreases to rationalize the valuation if the market’s P/B valuation of the stock is lower than the valuation of the stock’s essence (ROE). Conversely, the portfolios at the top left of the curved area move to the bottom right in the t + 1 quarter. This signifies that either the stock’s P/B in the next quarter decreases or the company’s ROE in the next quarter increases to rationalize the valuation if the market’s P/B valuation of the stock is higher than the valuation of the stock’s essence (ROE).

3. (3)

Figure 4 displays and indicates that the amplitude of horizontal motion (ROE) is much larger than the amplitude of the vertical motion (P/B) in the process of gathering to the bending area. It signifies that the market’s P/B valuation of the stock remains unchanged between the two quarters if they simply move to the left and right. However, the ROE of the stock drops or rises to the state consistent with the P/B valuation. This also indicates that the P/B in the t quarter has been properly reflected in advance and the market is efficient. If they simply move up and down, there are two possibilities: (a) it means that the market P/B overreacts to good or bad news, and then the market adjusts its P/B to a state consistent with the real implication of the information in the news; (b) it means that the market underreacts to the increase or decrease of ROE, and then the market adjusts its P/B to a state consistent with the ROE valuation. Judging from the fact that the magnitude of horizontal movement (ROE) is much larger than that of vertical movement (P/B), the market is quite efficient. However, there is still some inefficiency. Thus, its efficiency has not reached the perfect state yet.

4. (4)

The two sides of the curved area from the bottom left to the top right formed in the quarter t + 1 are asymmetrical. Also, the bottom right edge appears very tight and solid. Other than that, there are no portfolios outside. On the other hand, the top left edge is very loose with many portfolios scattered outside this side. This phenomenon is similar to the concept of efficient frontier revealed by the modern portfolio theory: a group of portfolios with the largest returns and least risks form the upper left efficient frontier. Rational investors will rush to purchase if there is a portfolio outside this frontier. This leads to the rising stock prices and the decline in return rates. Eventually, it will return to the efficient frontier.

5. (5)

If the ROE of the company is greater than the desired rate of return, the shareholders' equity will increase if the earnings are retained. In the meantime, the P/B is bound to fall if the stock price remains unchanged. On the contrary, it signifies that the increase of proportion of share price is greater than the increase of that of share equity if the P/B rises. The P/B of most portfolios at the lower right of the curved area rise in the t + 1 quarter if the ROE of the company is greater than desired rate of return. Thus, the stock price will inevitably rise and stocks of the portfolio gain higher returns. Conversely, the stocks of the portfolio at the upper left of the curved area obtain lower returns.

#### The ROE and P/B in the subsequent first and second quarter after the portfolios are formed

The ROE and P/B of the subsequent first and second quarters after the portfolios being formed are shown in Fig. 5. The circle end of the line segment is the position of the subsequent second quarter. The other end is the position of the subsequent first quarter. Figure 5 provides some important insights as follows.

1. (1)

The portfolios continue to gather in the curved area from the t + 1 quarter to the t + 2 quarter. However, the magnitude becomes rather small. This reflects that the market has responded to most of the ROE information in the t + 1 quarter. Also, there is little information that needs to be responded to in the t + 2 quarter.

2. (2)

The amplitude of the horizontal motion (ROE) is no longer much larger than that of the vertical motion (P/B) during the process of converging to the curved area. The adjustment no longer matches the market valuation with the great increase or decrease of the ROE of the company. However, it makes a slight adjustment in the market evaluation (P/B) and the ROE of the company in the t + 2 quarter.

3. (3)

The two sides of the curved area formed in the quarter t + 2 are asymmetrical. The lower right edge appears very tight and solid but there is still a tendency to move to the upper left. On the other hand, the upper left edge appears very loose and the moving direction scattered. This phenomenon indicates that the market has some insufficient response to prices of the "good" stocks. There is still some information that needs to be reflected on stock prices in the t + 2 quarter. However, the "bad" stocks have sufficient market responses. Also, there is little information that needs to be responded to the stock price in the t + 2 quarter.

#### The ROE and P/B in the subsequent second and third quarter after the portfolios are formed

The ROE and P/B of the subsequent second and third quarters after the portfolios being formed are shown in Fig. 6. The circle end of the line segment is the position of the subsequent third quarter. The other end is the position of the subsequent second quarter. Some significant findings are noticed from Fig. 6.

1. (1)

The portfolios continue to gather in the curved area from the t + 2 quarter to the t + 3 quarter but the magnitude becomes much smaller. This reflects that the market has responded to most of the ROE information in the t + 2 quarter already so there is little information that needs to be responded to in the t + 3 quarter.

2. (2)

However, in the portfolios that are farther from the curved area, the magnitude of changes in the t + 2 and t + 3 quarters is still quite significant. The general trend is gathering in the central area. It presents the characteristics of a two-dimensional mean reversion.

#### The ROE and P/B of the current and the subsequent eight quarters after the portfolios are formed

The ROE and P/B of the current and the subsequent eight quarters after the portfolios being formed are shown in Fig. 7. The circle end of the line segment is the position of the subsequent eighth quarter, and the other end is the position when the portfolios are formed. Figure 7 yields some important insights as follows.

1. (1)

The magnitude of changes of the portfolios in the upper right and the lower left, farther away from the central area, from quarter t + 1 to quarter t + 8 is in an obvious and overall trend of gathering to the central area. It shows the characteristics of a two-dimensional mean reversion.

2. (2)

The portfolios, which reach the curved area, do not continue to move significantly. This reflects that the curved area is a market equilibrium zone. Thus, the market gives a corresponding valuation (P/B) to stocks with good or bad company essence (ROE).

#### Smart frontier

Two important conclusions can be drawn from the above findings and analysis:

1. (1)

They tend to gather to the curved area in subsequent quarters when the portfolios are formed according to the company essence (ROE) and market valuation (P/B). The company essence and market valuation no longer change significantly after reaching this area. This signifies that the curved area is a market equilibrium area.

2. (2)

The two sides of the curved area are asymmetrical. The lower right edge appears very tight and solid but the upper left edge is very loose.

Therefore, we put forward the concept of the smart frontier, that is, the essence of the company (ROE) has a curved relationship with the valuation of the market (P/B). There are no investment portfolios at the bottom right of this curve. Hence, smart frontier can be regarded as the reasonable lower bound of P/B. This is because the P/B of the portfolio below this curve is lower than the lower bound of the reasonable P/B consistent with the valuation of the company's essence (ROE). This indicates that the stock price is undervalued. Therefore, smart investors in the market are to purchase these stocks. This is to result in rising stock prices on these stocks. Then, they return to the boundary. This relationship curve can be established by the Eq. (10), whose coefficients can be determined by the quantile regression analysis with market data.

Quantile regression analysis has been gradually and widely adopted in various research fields in recent years because it overcomes the shortcomings of the traditional regression analysis, which uses only the mean value as the dependent variable. Instead, quantile regression analysis adopts the median or another quantile values as the dependent variables of the regression models (Gowlland et al. 2009; Koenker and Hallock 2001; Mello and Perrelli 2003; Liao and Wang 2012; Uribe and Guillen 2020; Yeh and Liu 2020). For instance, it can construct a series of regression models for various quantiles such as 1/10, 2/10,…, 9/10.

The error function of the conventional regression analysis is displayed as Eq. (12) below.

$$E = \sum\limits_{i}^{{}} {\left( {Y_{i} - \hat{Y}_{i} } \right)^{2} }$$
(12)

Meanwhile, the error function of the quantile regression analysis is presented as Eq. (13) below.

$$E = \theta \cdot \sum\limits_{{Y_{i} \ge \hat{Y}_{i} }}^{{}} {\left| {Y_{i} - \hat{Y}_{i} } \right|} + (1 - \theta ) \cdot \sum\limits_{{Y_{i} < \hat{Y}_{i} }}^{{}} {\left| {Y_{i} - \hat{Y}_{i} } \right|}$$
(13)

where θ = the quantile. The value of θ is between 0 and 1. Y = the actual value of the dependent variable. $$\hat{Y} = \beta X$$ = the predicted value of the dependent variable. X = the vector of the independent variable. β = the coefficient vector to be estimated.

The error function to estimate the coefficient vector can be minimized if the value of the θ in the error function is chosen. For instance, if θ = 0.9, the weight of the error when the actual value is higher than the predicted value is 0.9, while the weight of the error when the actual value is lower than the predicted value is 0.1. Hence, the probabilities of the actual value being higher and being lower than the predicted value is balanced at 0.1 and 0.9. Therefore, the regression value is the ninth of the ten quantiles of the dependent variable. Thus, we can produce the predicted values for each quantile of the dependent variable if we employ several quantiles (for example, in the case of the ten quantiles, let θ = 0.1, 0.2…, 0.9) to construct the quantile regression models.

To find the smart frontier, the lower bound of the fair P/B, we utilize the market P/B in the eighth quarter after the portfolios are formed by ROE as the dependent variable and the current quarter ROE as the independent variable. We set the θ = 0.01; thus, the probability that the P/B in the eighth quarter after the portfolios are formed by ROE is lower than the predicted value is 0.01. Then, we employ quantile regression analysis to determine the coefficients in the valuation formula so we obtain Eq. (14) as displayed below.

$$P_{ \, 0} /B_{0} = k \cdot \left( {1 + ROE_{0} } \right)^{m} = 0.681 \cdot \left( {1 + ROE_{0} } \right)^{28.86}$$
(14)

By plotting the regression curve on the scatter diagram in Fig. 8, it is noticeable that after the formation of the portfolios, the ROE and the P/B in the subsequent quarters gradually converge to the regression curve and that there are no investment portfolios outside the frontier.

Figure 9 shows the moving paths of the ROE and P/B of the current quarter and the subsequent one, two, three, and eight quarters after the formation of the portfolios. It can be interpreted that the original portfolios at the bottom right corner of the curved area approach this regression curve and stay here. Meanwhile, the original portfolios at the upper left corner of the curved area approach this regression curve but do not cross this regression curve. This proves the existence of the smart frontier and the rationality of GVM theory.

### Data visualization: low-risk and high-risk stocks

To explore whether the concept of smart frontier can be applied to the low-risk and high-risk stocks, we divided the data set into two subsets with the minimum 20% and the maximum 20% of 250-day beta. The results are respectively exhibited in Figs. 10, 11, 12 and 13. The smart frontiers of Eq. (14) are also exhibited in these figures. These signify that the same smart frontier can be applied to the low-risk and high-risk stocks and indirectly verify that the theoretical closed form formula derived from GVM for a fair P/B is reasonable. Whether the risk is small or large, these figures also signify that most of them move in a horizontal way and close to the "smart frontier" after the portfolios are formed. It implies that risk is not an effective factor to trading strategy.

### Data visualization: small-cap and large-cap stocks

To explore whether the concept of the smart frontier can be applied to small-cap stocks and large-cap stocks, we divided the data set into two subsets of the minimum 20% and the maximum 20% of market value. The results are respectively shown in Figs. 14, 15, 16 and 17. The smart frontiers of Eq. (14) are also shown in these figures, which indicate that the same smart frontier can be applied to small-cap stocks and large-cap stocks. Implications from these figures include that:

1. (1)

Small-cap stocks react faster and are closer to the "smart frontier" than large-cap stocks after the portfolios are formed.

2. (2)

The direction of movement of large-cap stocks is relatively stable. Most of them move horizontally towards the "smart frontier". It implies that the market usually provides large-cap stocks with fairer valuation than small-cap stocks.

### Data visualization: small- and large-momentum stocks

To explore whether the concept of the smart frontier can be applied to the small and large momentum stocks, we divided the data set into two subsets: the minimum 20% and the maximum 20% of the t + 1 quarter return rate. The results are respectively shown in Figs. 18, 19, 20 and 21. The smart frontiers of Eq. (14) are also shown in these figures. These indicate that the same smart frontier can be applied to small and large momentum stocks. In addition, these figures signify that:

1. (1)

Stocks with lower returns in the t + 1 quarter show a downward trend in the t + 2 quarter. On the other hand, stocks with higher returns in the t + 1 quarter show a slightly upward trend in the t + 2 quarter. It implies that momentum is an effective factor to trading strategy.

2. (2)

Small momentum stocks move faster and closer to the "smart frontier" than large momentum stocks do after the portfolios are formed.

## Exploring and verifying return rate models with the application of data visualization

### Method

To build the return rate model, Eq. (11) can be rewritten as follows

$$1 + R = \lambda \cdot \left( {\frac{{B_{0} }}{{P_{0} }}} \right)^{b} \cdot \left( {1 + ROE_{0} } \right)^{c}$$
(15)

Take the log of both sides, Eq. (15) can be rewritten as follows

$$\ln (1 + R) = \ln \lambda + b \cdot \ln \left( {\frac{{B_{0} }}{{P_{0} }}} \right) + c \cdot \ln \left( {1 + ROE_{0} } \right)$$
(16)

Let the $$\ln (1 + R)$$ serve the dependent variable, $$\ln \left( {B_{0} /P_{0} } \right)$$ and $$\ln \left( {1 + ROE_{0} } \right)$$ as the independent variables, and $$\ln \lambda$$ as the constant term, then classical linear regression analysis can be employed to estimate the coefficients in the model.

### Data sources and data visualization

The relationships among ROE, P/B, and the rates of return are shown in Figs. 22, 23, 24, 25, 26, 27 and 28. Figures 22, 23, 24 and 25 are the 3D path diagram. Figures 26, 27 and 28 are their three-view drawings. These figures display the data of the t, t + 1, t + 2 quarter. Circle, diamond and x marks represent the data of t, t + 1 and t + 2 quarters respectively. X and Y axes represent ROE and P/B respectively. Z axis represents the cumulative quarterly rates of return. Thus, the Z value for the t, t + 1, and t + 2 data is respectively zero, the quarterly return rate in the t + 1 quarter, and the quarterly return rate in the t + 1 quarter plus that in the t + 2 quarter. Several salient features can be identified from these figures.

1. (1)

The data outside, inside, and close to the “smart frontier” has positive, negative, and around zero quarterly rates of return respectively in the t + 1 and the t + 2 quarter.

2. (2)

The three-view drawings show that the return rate in the subsequent first and second quarter relates proportionally to the ROE after the portfolios are formed. In the meantime, it relates inversely proportionally to the P/B.

To clearly explore the data set, Figure 29 shows the data in the t and t + 1 quarter. Also, Figure 30 shows the data in the t + 1 and t + 2 quarters. We may conclude that the data outside the “smart frontier” not only has positive quarterly return in the t + 1 quarter, but also has positive returns in the t + 2 quarter. Although the quarterly return rates in the t + 2 quarter are smaller than those in the t + 1 quarter, they are still certain. The farther the data in the t quarter away from the “smart frontier”, the larger the return rates in the t + 1 and t + 2 quarter.

Figures 31, 32, and 33 display the data of return rates in the t + 2 quarter and represent the ROE and P/B in the t, t + 1, and t + 2 quarter. Colors in red, yellow, green, and blue of Figs. 31, 32 and 33 represent the data with the quantile of return rate of 0–25%, 25–50%, 50–75% and 75–100%, respectively. These figures show that the data in the lower right area have higher return rates. Meanwhile, those at the upper left area have lower return rates and those alongside the “smart frontier” have median return rates. The ROE and the P/B in the t quarter can precisely predict the return rate of the t + 2 quarter. Furthermore, the smart frontier is able to accurately form a lower bound of P/B ratio at a specific ROE.

### Model configuration

Investment portfolios formed by the financial report of the t quarter cannot be invested at the beginning of the t + 1 quarter because there is always a lag period between the announcement date of the financial report and the end of the quarter represented by the report. Therefore, the quarterly return rate in the t + 2 quarter instead of the t + 1 is employed as the expected rates of return for the portfolios formed in the t quarter. Also, we utilize 100 data sets from the 100 portfolios with various combinations of the ROE and the P/B and their corresponding quarterly return rate in the t + 2 quarter.

The above 100 data sets of investment portfolios are plotted on the three-dimensional scatter diagrams as shown in Figs. 34 and 35, where the dots are the three-dimensional (X, Y, Z) positions of the portfolios formed in the t quarter. X axis stands for ROE, Y axis stands for P/B in the t quarter, and Z stands for the rate of return in the subsequent t + 2 quarter. The other end of the line segment is projected to the Z = 0 plane. Therefore, the rate of return for the subsequent t + 2 quarter is negative when the dot is at the lower end of the line segment.

The regression coefficients of Eq. (16) can be obtained by adopting linear regression analysis on the data sets of these 100 investment portfolios. The results are exhibited in Table 1. The adjusted coefficient of determination is 0.548. Figure 36 shows actual and predicted values of the quarterly return rate. It indicates that the model cannot accurately predict some outlier data. Figures 37 and 38 are produced and presented to explore the above issue. They show the surface of the predicted value and indicate that the model underestimates the return rates when the ROE is relatively low.

The above-mentioned model underestimates the return rates when the ROE is relatively low. A modified model is proposed in order to improve modelling accuracy and is presented below.

$$R = \lambda \cdot \left( {\frac{{B_{0} }}{{P_{0} }}} \right)^{b} \cdot \left( {1 + ROE_{0}^{*} } \right)^{c} - 1$$
(17)

When $$ROE_{0}^{{}} > ROE_{UB}^{{}} {\text{ Then}}\,\,ROE_{0}^{*} = ROE_{UB}^{{}}$$.

When $$ROE_{0}^{{}} < ROE_{LB}^{{}} {\text{ Then}}\,\,ROE_{0}^{*} = ROE_{LB}^{{}}$$.

Otherwise $$ROE_{0}^{*} = ROE_{0}^{{}}$$.

To build the modified model, two-phase algorithm is employed as follows.

1. (1)

Use an evolutionary algorithm (Kramer 2017) to estimate the upper bound and lower bound of ROE, and coefficientsλ, b, and c simultaneously.

2. (2)

Fix the upper bound and lower bound of ROE, and employ the classical regression analysis to estimate the coefficients λ, b, and c.

Table 2 shows the upper bound and lower bound of ROE, and coefficients λ, b, and c. Its adjusted coefficient of determination increases from 0.548 to 0.709. Some profound findings from this table indicate that the coefficients λ (1.020) and c (1.05) are both close to 1.0.

According to Eq. (11), when λ, c, and $$B_{0} /P_{0}$$ are equal to 1.0, then the expected return rate is the same as Return on Equity (ROE), which is considered as the measure of a corporation's profitability in relation to stockholders’ equity. Because $$B_{0} /P_{0}$$ = 1 means that the stock price equals stockholders’ equity per share, under this condition it is reasonable to infer that the expected return rate is the same as ROE for a long run. The above findings are consistent with the inference.

Figure 39 shows actual and predicted values of the quarterly return rate. It indicates that not only does the outlier data disappear but also the accuracy of the model also improves in Fig. 39. Figure 40 and 41 show the surface of the predicted value and signify that the error is randomly distributed throughout the data domain. The results show that it is necessary to employ the upper bound and lower bound of the current ROE to build the accurate model on the expected rates of return. The possible reason is that too low and too high ROE cannot persevere, which conflicts with the most important assumption of GVM that the ROE follows the principle of mean reversion.

## Conclusions

Growth Value Model (GVM) proposed theoretical closed form formulas for fair stock price and expected rates of return. Data visualization refers to techniques used to communicate insights from data through visual representation. With interactive visualization, users can take the concept a step further by using technology to drill down into charts and graphs for more details. It changes the way how the data is to be presented and how it’s processed. Therefore, it can yield insights which traditional descriptive statistics cannot. This article uses visualization techniques to help explore GVM intuitively. We conclude that:

1. 1.

Exploring the fair P/B ratio with the application of ROE-P/B 2D dynamical process visualization

1. (1)

The data points of the subsequent quarters are clearly gathered in the curved area from the lower left to the upper right on the ROE-P/B diagram after the portfolios are formed. This signifies that the market gives higher P/B valuations to stocks with higher ROE. The market is quite efficient. However, there are still some inefficiencies.

2. (2)

The two sides of the curved area are asymmetrical. The lower right edge appears very tight and solid. There are no portfolios outside this edge and there is a trend of moving to the upper left. However, the upper left edge appears very loose with many portfolios scattered outside this edge. Also, the direction of the movement is scattered.

3. (3)

We put forward the concept of the smart frontier. Thus, there is a curvilinear relationship between the essence of the company (ROE) and the valuation of the market (P/B). There are no investment portfolios at the lower right of this curve. Therefore, the smart frontier can be regarded as the reasonable lower bound of P/B at a specific ROE. This relationship curve can be obtained by the formula derived from GVM with the quantile regression analysis and market data. The moving paths of the ROE and P/B in the current quarter and the subsequent quarters show that the portfolios at the lower right of the curved area approaches this regression curve and stagnate here after the portfolios are formed. This proves the existence of the smart frontier and the rationality of GVM theory.

4. (4)

The stocks with various characteristics such as: risk, market value, and momentum, have the same trend where most of them move in the direction close to the "smart frontier".

1. 2.

Exploring the expected rates of return with the application of ROE-P/B-Return 3D dynamical process visualization

1. (1)

The return rate in the subsequent first and second quarter is directly proportional to the ROE. Also, it is inversely proportional to the P/B.

2. (2)

The data outside the “smart frontier” has a positive quarterly return rate not only in the t + 1 quarter but also in the t + 2 quarter. Although the quarterly return rates in the t + 2 quarter are smaller than those in the t + 1 quarter, they are still certain. The farther the data in the t quarter away from the “smart frontier”, the larger the return rates in the t + 1 and t + 2 quarter. Therefore, it implies that smart investors should purchase these undervalued stocks until the stock prices rise and return to the boundary.

3. (3)

The original GVM model cannot accurately predict some outlier data. It underestimates the return rates when the ROE is relatively low. Not only does the outlier data disappear but also the accuracy of the model also improves by adding the upper bound and lower bound of ROE into the original GVM. The possible reason is that too low and too high ROE cannot persevere, which conflicts with the most important assumption of GVM that the ROE follows the principle of mean reversion.

Suggestions for future research include.

1. (1)

To examine whether the findings can help investors to improve their investing performance, an empirical study evaluating the out-of-sample risk-adjusted returns of different portfolios sorted by the expected rate of return model can be conducted.

2. (2)

To explore whether the findings of the dynamic behavior of smart frontier are robust and general, other emerging or developed markets can be examined.

## Availability of data and materials

The dataset on which the conclusions of the manuscript rely is a secondary data and it will be made available upon request.

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### Contributions

I-CY conducted the project and wrote the paper. Y-CL participated in the joint revision of the paper. Both the authors read and approved the final manuscript.

### Corresponding author

Correspondence to I-Cheng Yeh.

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Yeh, IC., Liu, YC. Exploring the growth value equity valuation model with data visualization. Financ Innov 9, 2 (2023). https://doi.org/10.1186/s40854-022-00400-2